Problem: Rewrite the function by completing the square. $h(x)= 2 x^{2} +11 x +15$ $h(x)=$
Explanation: $\begin{aligned} h(x)&=2 x^2 +11 x +15 \\\\ &=2 \left(x^2 +\dfrac{11}{2} x\right) +15 \end{aligned}$ Now we want to complete $x^2 +\dfrac{11}{2} x$ into a perfect square. To do that, we should add $\left(\dfrac{{\frac{11}{2}}}{2}\right)^2={\dfrac{121}{16}}$ to it: $x^2{+\dfrac{11}{2}}x+{\dfrac{121}{16}}=\left(x +\dfrac{11}{4}\right)^2$ We add ${\dfrac{121}{16}}$ inside the parentheses, and subtract ${2}\cdot{\dfrac{121}{16}}$ outside them, to keep the expression equivalent. $\begin{aligned} &\phantom{=}{2} \left(x^2 +\dfrac{11}{2} x\right) +15 \\\\ &={2}\left(x^2 +\dfrac{11}{2} x+{\dfrac{121}{16}}\right) +15 -{2}\cdot{\dfrac{121}{16}} \\\\ &=2 \left(x +\dfrac{11}{4}\right)^2 +15 -\dfrac{121}{8} \\\\ &=2 \left(x +\dfrac{11}{4}\right)^2 -\dfrac{1}{8} \end{aligned}$ In conclusion, the function after completing the square is written as: $h(x)=2 \left(x +\dfrac{11}{4}\right)^2 -\dfrac{1}{8}$